I am not sure how to show immediately that it is irreducible; however here is another idea.
We know that:
But it is easy to show each of the intermediary steps is a degree 2 extension with irreducible minimal polynomials all irreducible by Eisenstein with p = 5, 3, 2 respectively. So that means the total extension is:
Since you have found a monic polynomial of degree 8 that has this as a root it must be irreducible. Kind of roundabout, but hope it helps.